Subject: mathematics question
Date: Sun, 2 May 1999 01:47:43 +0100
A shape preserves its shape if a rotation, translation or scaling is performed on it. Are these the only continuous transformations which have this property? These transformations if performed on the parts and then summed have the same effect as the transformation being applied to the whole; are these linear transformations? Who, and what area of mathematics has classified all transformations of this type completely?
thanks for your time,
The shape-preserving transformations in the plane:
The shape-preserving transformations of 3-space.
REFERENCE. I have taken the terminology from H.S.M. Coxeter's INTRODUCTION TO GEOMETRY (CHAPTERS 3, 5, AND 7). There are many textbooks that deal with isometries; many high-school texts have a chapter on the topic.
HISTORY. Geometers have always thought dynamically, using transformations long before the concepts were carefully defined. For example, some of Euclid's proofs (300 BC) rely on isometries. In 1872 the German mathematician Felix Klein proposed that transformations be used as the basis for geometry, with Euclid's axioms being supplemented by transformation axioms. The modern interest in transformation groups can be traced back to him, although he would probably admit that none of his ideas were original.
Finally, YES, all isometries and similarities are linear transformations: They can be described using matrices acting on vector spaces.
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